\newproblem{lay:1_5_39}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.5.39}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $A$ be a $m\times n$ matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors with the property that $A\mathbf{v}=\mathbf{0}$ and
	$A\mathbf{w}=\mathbf{0}$. Explain why $A(\mathbf{v}+\mathbf{w})=\mathbf{0}$. Then, explain why $A(c\mathbf{v}+d\mathbf{w})=\mathbf{0}$ for
	each pair of scalars $c$ and $d$.
}
{
  % Solution
	We know that
	\begin{center}
		$A\mathbf{v}=\mathbf{0}$ \\
		$A\mathbf{w}=\mathbf{0}$ \\
	\end{center}
	Adding both equations
	\begin{center}
		$A(\mathbf{v}+\mathbf{w})=\mathbf{0}$
	\end{center}
	As stated by the problem. For showing that $A(c\mathbf{v}+d\mathbf{w})=\mathbf{0}$ we may follow a different strategy
	\begin{center}
		$\begin{array}{rcll}
		   A(c\mathbf{v}+d\mathbf{w})&=&A(c\mathbf{v})+A(d\mathbf{w}) & \text{By distributivity of matrix multiplication} \\
		                             &=&c(A\mathbf{v})+d(A\mathbf{w}) & \text{By scalar product property of matrix multiplication} \\
		                             &=&c(\mathbf{0})+d(\mathbf{0})   & \text{By definition of }\mathbf{v}\text{ and }\mathbf{w}\\
		                             &=&\mathbf{0}                    &
		 \end{array}$
	\end{center}}
\useproblem{lay:1_5_39}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
